cheesemonkey wonders

cheesemonkey wonders

Sunday, August 12, 2018

A Course in Thinking

This blog post is also a session at Sam Shah's The Virtual Conference of Mathematical Flavors 


In a conference on flavors of mathematical teaching and learning, you could be forgiven for expecting every session to address some version of the age-old arguments about whether there are math people and non-math people, about whether it is better to have a growth mindset or a fixed mindset, or about whether mathematics is the most beautiful of all the disciplines we teach in school.

Which is probably why I feel so hesitant to confess my dirty little secret. In my classroom, I teach mathematics as one of the humanities.

The fact is that math is a human activity. If you are human, you cannot escape it. And what I have come to value the most about the opportunity to teach mathematics is that it has become one of the most pivotal ways in which we transmit the culture and values we cherish the most. For me, some of those values include respect, communication, empathy, understanding, persuasion, civil disagreement, persistence, deep listening, reassessing, and changing one’s mind. Math class gives me an opportunity to share all of these aspects of being human and living together in human culture.

So I’ve rewritten the introduction to my course syllabus to emphasize some of the things about which I feel most strongly — and which I believe are the most powerful and important things I have to share with my students over the course of the next year.

I listen to hear your thoughts.
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This is a course about thinking.

You are here to learn how to think better and to use your thinking to accomplish things in the world.

The essence of thinking is sense-making. To make sense of things, you have to understand them, which means you have to want to understand them. One of our mottos in this class is, You gotta wanna.  This is as important in mathematics as in everything else.

So everything in this class is about making sense of things. In mathematics — as in life — we mostly make sense of problems. If you do not yet know that life presents a steady stream of problems to be solved, you will soon. 

In this class, we happen to use mathematics  as a ground for thinking. I will tell you a secret up front: I don’t actually care if you  ever “use” this stuff ever again or not... so please don’t waste time asking me that question. It is a boring and senseless question. What I care about passionately is that you learn how to think and communicate at a more advanced level than you are capable of right now. 

And that is what we are going to work on.

Thinking better is a set of skills you can actually learn and use at this school. It is the appropriate focus of math class.

Since you are going to be thinking for the rest of your lives, you are going to need to make sense of things you don’t initially understand. And then you’re going to have to persuade other people that your thinking is right. So your goal in this course should be to grow as an active sense-maker who is skilled in using these tools of thinking.

You should also learn to treat your thinking with respect. The mind is a muscle, and this school is a place where we work to strengthen our thinking muscles. That means we need to develop strength, flexibility, and endurance in our thinking — in other words, you need to become a strong thinker, a flexible thinker, and a persistent thinker. You also need to become a good collaborator, which means you need to become a better listener.


While there are no guarantees, I can promise you that if you focus on these goals here, you will do well in this class, and these skills will carry you very far in your life.

Sunday, July 29, 2018

TMC 18 Recap: a lesson from the Rock and Roll Hall of Fame: Find what you need. Refuse to be stopped.

On my last day in Cleveland, I went to the Rock and Roll Hall of Fame. This was my one assignment from my husband David, a long-time jazz, rock, and world music radio DJ who has not yet had his own reason to visit Cleveland.

At the end of the third floor Hall of Fame exhibit is the "Power of Rock Experience," a small, stadium-style theater with a big-screen, state-of-the-art showing of Jonathan Demme's highlight film of Hall of Fame Induction concerts. Strobe, lighting effects, and fog machines give you a powerful, close-up concert experience, even though you are in a small, stadium-style theater in Cleveland.

What stayed with me was the climax of the film, the joyful, posthumous concert tribute to George Harrison, in which Tom Petty, Prince, George's son Dhani, and a number of other amazingly famous RRHOF inductees gave their own burning-down-the-house tribute version of While My Guitar Gently Weeps. It was a killer tribute to a killer song that has meant a lot to me in my own life. Prince's guitar solo, tearing up Eric Clapton's original version on the Beatles' recording, blew my mind.

When I called my husband later that night to tell him about it, David said, "Isn't that the one where Prince throws his guitar into the air and it never comes down?"

I laughed and said, "Yep, that was it."

That's when it hit me—the lesson I have taken away from my seven years of attending Twitter Math Camp: Find what you need. Refuse to be stopped.

The story of that song, as I understand it, was and remains amazing to me. George was frustrated by his inability to get his songs onto the Beatles' albums. He felt like he couldn't get John and Paul to pay attention to this song and he vented to everybody he knew. Finally, after venting to his good friend Eric Clapton, Eric came to their next recording session to support George. He sat in with them, played the guitar solos that made them really hear the power of the song, and the rest was history.

Seven years ago, when we conceived and held the first Twitter Math Camp in St. Louis, nobody noticed us. Nobody cared. We were 39 North American math teachers who had a yearning for community. A couple people came from Canada. One person came from Amman, Jordan. I was the only person who came from California. In fact, we had more participants from Mississippi than we had from California.

And we created something powerful, something we needed.

It was a stone soup effort, and it still is. What people may not know is that this is something that was created out of thin air. A core group of about 15 of us agreed to provide the core structure. There was no organization, no staff at that time, except for Jason Henry, who was about to join an effort he didn't even know much about.

People just showed up and brought what they had.

And what they had to share was amazing.
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I learned a lot from that experience, but the most important thing I learned was what I consider to be the essence of Twitter Math Camp:
Complain and vent if you must, but then find what you need and refuse to be stopped.
Know that there is going to be loneliness, there is going to be heartache, and there is going to be risk.

But also know that there is going to be a miracle.

When the Beatles started, they were four dudes from Liverpool who wanted to play rock and roll so much that they spent their adolescences doing what they loved, making music and playing for people in dark, dank basement clubs in Germany.

They did this over and over and over — for years.

They had no idea that what they were going to become was THE BEATLES. They had found what they loved, and they committed themselves to honing their craft. They refused to be stopped.

By the time they got their first recording contract, they had gotten so good they could bang an amazing album out in just a few takes.

They kept doing it even after it became tougher to do.

Eventually, they gave up giving live concerts altogether. The thing they had loved the most had become dangerous and damn near impossible. But they never stopped loving it. In the last "public" concert they gave together, on the roof of Apple Records in London, they played to the sky and to anybody within earshot who could hear them. Fortunately for us, one of their managers had the foresight to film this event, which occurred not long before they broke up forever.

But for me, sitting in the darkened theater, watching Tom Petty and Prince and Dhani Harrison bring down the house with this performance of a song that almost never saw the light of day, there was joy in remembering how this same spirit of determination brought me this annual retreat/conference/event that has become so dear to me.

Find what you need. Refuse to be stopped.

Here are some of the other important secrets I have learned from seven years of TMC.

Get busy and recognize belonging as blessing. Find what you need and need to share. Refine your own craft. Trust that people like you need you and are searching for you. If one group is unable to see and value what you need yet, refuse to be stopped. Continue under all circumstances and keep searching for your people. If you haven't found them yet, keep searching.

We were nobody. I'm still not sure how we became somebody. I remember The Great Facebook Friending of Winter Break 2011, when I was certain that I was going to wake up and discover why you should never friend and meet people you have only ever met on the internet. I was sure I was going to have to go into Facebook Witness Protection. But it turned out OK. In fact, I made a number of lifelong friends that way.

So keep following and keep friending. No risk, no reward. Remember that the people you are looking for are also looking for you.

Practice Gratitude. Gratitude is a giant, holy yes that I keep saying over and over and over. When I found someone who was generous with their blog and their tweeted advice and their encouragement, I said thank you. My way of saying thank you in those earliest days was to say "thank you" over and over but when somebody was inconceivably generous with me, I knitted them a small, stellated dodecahedron and mailed it to them. I didn't ask. I didn't promise. I just did it. I did this thirteen times.

I've been just as blessed — and surprised — to receive things for my own generosity. During a dark time, Tina Cardone crocheted me a unicorn that now sits on my desk. @veganmathbeagle crocheted me an otter. @caseymcteach mailed me a monkey lanyard. Kristin Fouss gave me a Fiona the baby hippo t-shirt. And other treasures too numerous to elaborate here.

The point is, the practice of gratitude is a big part of the essence of Twitter Math Camp. It's one of the invisible threads that bind us together.

If you need this kind of connection, keep searching and keep practicing gratitude. It will come back to you many times over if you let it.

Be unconditionally constructive. This is especially important on Twitter itself. With all the negativity in our world right now, I think the best thing we can each do is to contribute unconditional positivity wherever we can. That's a big part of being a teacher, to be sure, but it's also a critical part of being a digital citizen.I am constantly trying to remember to ask myself, What am I contributing right now?

The Power of 'Yet.' Refuse to be discouraged. If people haven't come around to your point of view yet, recognize that you just haven't reached them or persuaded them yet. Keep going. Keep growing, Keep working. If you're right, you'll convince people eventually.

Make what you need. While #tmcjealousycamp is fun and funny, it's beside the point. What is important is to figure out what you need and push forward and make it. That is the power of un-conferences and salons and all the other forms of community. If you are committed to finding your community of math teachers, that's it. Don't let anybody stop you. Figure out what you need and create it. You'll be astonished at the support you get back from the Universe.

Friday, April 6, 2018

HERESY WARNING: Breaking through on quadratic function analysis & graph sketching

Over the last two days, I've had a few important breakthroughs with discouraged Algebra 1 learners about quadratic functions and their graphs. I wanted to document this for myself before I have a chance to forget about it for next year.

If you have rigid beliefs about the only ways for students to approach quadratic functions, analysis, and graphs, then this post is definitely not for you.

Consider yourselves warned.

My most discouraged Algebra 1 learners are extremely gifted kids, but this year's crop are definitely dreamers, not stalkers. They need to really marinate in something for a long time before it takes root in their minds. They are factoring warriors, but the quadratic formula and complicated answers can be really daunting for them.

The connection I have wanted all of my students to make between quadratic functions and their graphs is this: even when a quadratic doesn't have neat, simple, integer answers, it still has a number of neat, simple aspects that they can grab hold of. There will always be an axis of symmetry. There will always be a vertex.

What I have discovered — or rather, what they have been teaching me all this week — is that if you approach a quadratic function from the understanding that you already know how to find the AOS and the vertex, you can make a TON of important discoveries and understandings about its graph and the many properties of the function that are important to understand.

What they taught me today is that they know how to use the axis of symmetry formula like a key in a lock. They can identify a, b, and c in a quadratic, and they understand that they can find and graph the AOS equation quickly and fearlessly. Then they can plug in the value they found for the axis of symmetry to identify the vertex of the parabola.

From there, it is a simple matter to find some more points for your sketch and to identify their mirror reflections across the axis of symmetry.

The beauty of their method is that it makes it easy for them to develop a meaningful conjecture about whether or not a quadratic function has any zeros.

If the parabola is floating above the x-axis, then they can use the QF to confirm their hunch that there are no real-number zeros to the function. Likewise if the parabola is submerged below the x-axis.

I love that they naturally figured out today what it means for the vertex to be a maximum or a minimum.

And I especially love the fact that they made these discoveries themselves.

They still don't understand how to complete the square or use the quadratic formula to blast through problem after problem to find complicated zeros of non-trivial quadratics.

But this feels less important to me than the fact that they have made important connections and developed their own methods for investigating quadratic functions. And it has been an important reminder to me to design learning experiences that empower them to make these connections and discoveries for themselves.

Sunday, January 28, 2018

The Centrality of the Cross-Racial Therapeutic Alliance in My Classroom


As part of our school's five-year program of equity transformation, we the staff engage in monthly two-hour seminars focused on issues of equity and diversity. As someone who is leading (and learning to lead) equity transformation in my very large school, I live on both sides of the fence in this process. As a peer, I am often a witness to frustration and discouragement that naturally arise in an inherently vexed and vexing process, since we are only in year two of the process. And as someone on the leadership team, I am often tasked with encouraging greater engagement in a program whose theory of action is I sometimes have issues of practice with.

Nevertheless, I wanted to explore my own understanding of an important part of this month's reading included an article titled, "Racial Microaggressions in Everyday Life: Implications for Clinical Practice," by Derald Wing Sue et al (May—June 2007, American Psychologist). 

Most of my colleagues' responses to this piece have centered on the numbingly detailed taxonomies the authors present about forms of microaggressions, microassaults, and microinvalidations. This part of the article felt deeply Aristotelian to me — busily categorizing, classifying, analyzing, exemplifying, and naming these forms of everyday racist aggressions that occur every day everywhere in our country.

What has struck me most in this article — and what has stayed with me — was the discussion toward the end of the piece about the centrality of the therapeutic alliance within cross-racial dyads. This hit home for me because it is what I experience as the most central part of my own work in my classroom with my students of color, so I noticed myself sitting up straighter and taking much better notes when I was reading this section.

A lot of research on the effectiveness (or ineffectiveness) of the therapeutic process has zeroed in on the therapeutic alliance as the most essential determinant of success in personal transformation. In therapy, this means that the client must perceive the therapist himself or herself as being both deeply trustworthy and expert in the process of transformation. It seems to me that this same relationship exists within the classroom. The bond I have to form with my students requires that they perceive me as both worthy of their trust and expert in the process of transformation.

In addition to the client's perception of the therapist, the client must also have a very specific perception of him- or herself within the therapeutic context — namely, they must perceive that they themselves are being deeply understood and received in a relationship of unconditional positive regard. This too is a critical part of my own culturally responsive pedagogy. My students must perceive themselves as being welcomed to "come as you are" in my classroom because I know that this is the only thing that will make them feel safe enough to become vulnerable enough to engage in the transformative mathematical identity work that we do in my classes.

A huge part of my own practice of trustworthiness with my students  is about being open to accepting responsibility for harms that I cause (or perpetuate) and for taking corrective actions to repair the harms and clean up my side of the street in our relationships. This becomes even more important as a prerequisite for cultural competence, because I am sure there are a zillion unconscious ways big and small in which I, as a White teacher microassault, microinsult, or microinvalidate my students of color while at the same time using every tool in my power to love and support them. Being in community is like being in any relationship. You can never fully avoid running into conflict, but you can learn better ways to repair the local harms that arise so that the underlying fabric of the relationship remains intact for the benefit of all parties.

Because most helping professionals, including teachers, experience themselves as fair and decent people who are incapable of causing deliberate racist harm toward our students of color, it can be difficult to see microaggressions as they are happening because they are so often invisible to authority figures who are accustomed to swimming in a soup of white privilege. Yet I must continue to practice noticing my own white racial identity in order to develop this capacity in my conscious mind and social practice.

For me, the most urgent idea in this article was the fact that when racial microaggressions continue to occur within a cross-racial therapeutic context without being noticed, named, and addressed, clients of color are at risk of not continuing in their own work of inner development. To put it in the context of my classroom, if students perceive that I perceive myself to have no responsibility for noticing, naming, and addressing racial injustices at the micro- level within our relationship, then they are more likely to become discouraged and give up.

I don't know where all this thinking is going to take me. I only know that framing the cross-racial teaching and learning relationship as a therapeutic alliance feels transformative to me. And that means I need to keep investigating my role in it.

Sunday, September 17, 2017

Conversation With Grace: Why Strive For Diversity?

Ever since I read Grace's post, I've been struggling.

I have nothing but bad, inadequate answers to this question. And since its premise lies at the heart of my every day's work, I have felt stuck as I struggle to find a better answer.

I still have only terrible answers, but I need to get unstuck so I can get going again. So I'm writing this post as a chance to stick a pin in the best bad answers I have right now.

1. tikkun olam
Tikkun olam is the central assignment in the spiritual curriculum I grew up in. It is the requirement that we work actively to heal the world. When I was a child in the 60s, our rabbi was an active figure in the civil rights movement — a friend and colleague of Dr. King's, one of the Freedom Riders, and "the most arrested rabbi in America." As Dogen Zenji, the founder of the Zen lineage said, "When you walk in the mist, you get wet." Spending day after day, week after week in this community, I walked in the mist. I got wet. I came to believe in the power of tikkun olam to make America a more just place, even if it was still massively imperfect. Tikkun olam also taught me that if we as a society have the possibility of providing quality education and access to power for some, then we need to make sure that we make that possibility a reality. Our society and our world are a mess right now, so there's no time to lose. And the only way to heal the world is to actively work toward healing the world. I have to accept that I may be doing it wrong or badly, but in making positive effort for the good each day, I am doing the best that I can. And the more people I reach, the better. Because I never know who will trigger...

2. the hundredth monkey principle
In the 1960s and 70s, the hundredth monkey principle became an archetypal myth and story for me about how critical mass seems to come about in human society. The story has been used and recycled in many different contexts, but what strikes me as important is this: the story provides a useful metaphor for me about what happens as we share good ideas and carry them with us. For me, the big idea is that we can have no idea where or with whom the tipping point or critical mass will happen. So sharing widely — and diversely — is important to me because I have no idea who will be that "hundredth monkey." Identifying the hundredth monkey is above my pay grade. My job is to share best practices and wisdom so that it reaches the hundredth monkey and we can achieve some degree of tikkun olam sooner rather than later. The more people I reach, the better.

3. diversifying the power structure
I was grateful that Julie Wright shared this clip of Michelle Obama sharing her beliefs about the power of diversifying the power structure after watching the film Hidden Figures. Michelle and I were at Princeton at the same time, and we both benefited enormously from its diversity initiatives. I was in only the tenth class of women at Princeton, and Michelle was two years behind me. The education we received there was transformative, but it was also transformative to live in community with others from such dramatically differing backgrounds from around the world. It was my first experience of gifted education, and it changed me. It changed a lot of us. Several years later, I was privileged to be a part of a conversation at a conference in which Henry Louis Gates, Jr. framed the need to diversify the power structure and the power elite if we want true transformation, and I believe that for better or for worse, he is right. If we want to transform our society, we need to transform its power structure, and for that, we need to diversify our classrooms. As I've gotten older, I've gotten less apologetic for my elitist tactics here. I have less time to waste, and I want a better quality of leaders across the board in our pipelines.

4. Yes, it is better for me.
This is the argument I feel most ashamed of, but I have to be ruthlessly honest about my own motives. I hate it when I have to admit that I am ruled by enlightened self-interest, but it's a reality in economics and it's often a reality of my own life. It has nothing to do with pity for others and everything to do with what makes my own experience in the world I live in better. I love exploring the world around me, and that means I love experiencing viewpoints and perspectives that are different from my own. In my previous career in high tech, I experienced how blind spots take over when everybody at the table looks the same. In education, I see how the wealthy and powerful push idiotic agendas when they only listen to other wealthy and powerful people.

So yes, I work for greater anti-racist diversity in my educational context because it is better and healthier and saner for me.

I have a strong suspicion it is also better for others, but by the principles of tikkun olam, I am restricted to cleaning things up on my own side of the street.

I would love for us to have ever more diverse TMCs, but I am also bound to respect the differing boundaries, needs, and wants of the people who would make those things more "diverse" for people like me. I am bound to respect that for those who spend their school year isolated within an oppressively dominant culture, it may not feel optimal to them to spend their summer resources attending a PD conference that is still largely populated by teachers from the dominant cultures. That puts the onus on me and others like me to strive to make TMC a worthwhile and valuable choice for them to make. But it's critical to recognize and respect that those who would make a TMC "more diverse" place for me have their own needs, wants, and priorities. I need to make sure I am not trying to manipulate others into serving my own needs, wants and priorities.

I cannot bear to #pushsend, but I also can't bear to feel stuck any more.

I want to get back to giving my wild and imperfectly diverse classes the best educational opportunities I can give them, so I'm going to #pushsend anyway.  God help me.

Thursday, August 24, 2017

Why does this feel so dangerous...?

Maybe it's just the cold medicine speaking (I am battling a wicked cold), but today in 6th block Algebra 1, as we were working on Exeter problems, I ditched my usual, "safe" piece of lecture and went went for the jugular.

I wanted to give them the real stuff — not the party line position.

I advocated for radical, student-centered sense-making.





WHY DOES THIS FEEL SO DANGEROUS?????

Saturday, July 22, 2017

"You are what you seek," the wise one said.

Once upon a time, some time late in 2011, there were some lonely, kooky, determined math teachers trying to get better.

They searched blogs and joined Twitter in their quests, and eventually they found some like-minded spirits on the internets who were also questing.

In 2012, forty of us decided to meet in person in St. Louis and hold our own conference. Nobody outside that first group (besides Fawn, who hosted #TwitterJealousyCamp) knew or much cared about what we were doing. We were doing it because we wanted to do it. Period.

FUN FACT: Out of forty attendees at the first TMC, I was the one and only attendee from California. In fact, we had more attendees from Mississippi than from California.

It wasn't perfect, but it was real — and that kindled a spark. What made it magical was the fact that people showed up and brought their A game. I learned something amazing from every single person at that conference.

So if you attending TMC for the first time this year, please temper your freaking out with the knowledge that we started this thing because we were looking for YOU. We are STILL looking for you.

Before TMC, I always think of one of my favorite quotes from the great Jungian psychoanalyst and storyteller Clarissa Pinkola Estes:
  Even though there are negative aspects to it, the wild psyche can endure exile. It makes us yearn that much more to free our own true nature and causes us to long for a culture that goes with it. Even this yearning, this longing makes a person go on. It makes a [person] go on looking, and if she cannot find the culture that encourages her, then she usually decides to construct it herself. And that is good, for if she builds it, others who have been looking for a long time will mysteriously arrive one day enthusiastically proclaiming that they have been looking for this all along.